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BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR nonlinear evolution equations

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Acta Mathematica Scientia 2014年第4期34卷 1271-1286页

作者：彭艳 School of Mathematics and Statistics Central China Normal University

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as... 详细信息

关键词： nonlinear evolution equations vanishing diffusion limit convergence rates boundary layer BL-thiekness

STUDY ON EXACT ANALYTICAL SOLUTIONS FOR TWO SYSTEMS OF nonlinear evolution equations

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Applied Mathematics and Mechanics(English Edition) 2001年第8期22卷 925-934页

作者： YAN Zhen-ya(闫振亚) ZHANG Hong-qing(张鸿庆) Department of Applied Mathematics Dalian University of Technology Dalian 116024 P R China Department of Applied Mathematics Dalian University of Technology Dalian 116024 P R China

The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.

关键词： nonlinear evolution equations improved homogeneous balance method exact analytical solution solitary wave solution rational solution

DECAY OF GLOBAL SOLUTIONS OF TWO nonlinear evolution equations

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Acta Mathematicae Applicatae Sinica 1997年第1期13卷 23-32页

作者： GUO BOLING ZHANG LINGHAI(Institute of Applied Physics and Computational Mathematics, Beijing 100088, China) Institute of Applied Physics and Computational Mathematics Beijing China

We study the decay of solutions of two nonlinear evolution equations: the Benjamin-OnoBurgers and the Schrodinger-Burgers equations. We establish sharp rates of L2 decay of global solutions to these problems, with initial data Uo(x)∈L1∩L2. The decay results of the solutions follow from the a priori L2 integral estimstes and the Fourier transform. The standard argument relies on a technique that involves the splitting of the phase space into two time-dependent subdomains.

关键词： Decay estimates nonlinear evolution equations

A deep learning method for solving third-order nonlinear evolution equations

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Communications in Theoretical Physics 2020年第11期72卷 19-29页

作者：李军陈勇 Shanghai Key Laboratory of Trustworthy Computing East China Normal UniversityShanghai200062China School of Mathematical Sciences Shanghai Key Laboratory of PMMPShanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghai200062China College of Mathematics and Systems Science Shandong University of Science and TechnologyQingdao266590China Department of Physics Zhejiang Normal UniversityJinhua321004China

It has still been difficult to solve nonlinear evolution equations *** this paper,we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data ***,the model uses a deep neural network constrained with given governing equations to try to learn all optimal *** particular,numerical experiments on several third-order nonlinear evolution equations,including the Korteweg-de Vries(KdV)equation,modified KdV equation,KdV-Burgers equation and Sharma-Tasso-Olver equation,demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.

关键词： deep learning nonlinear evolution equations soliton interaction nonlinear dynamics

Using a New Auxiliary Equation to Construct Abundant Solutions for nonlinear evolution equations

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Journal of Applied Mathematics and Physics 2021年第12期9卷 3155-3164页

作者： Yifan Liu Guojiang Wu United Christian Academy Civic Center Drive Rancho Cucamonga USA Institute of Plasma Physics Hefei Institutes of Physical Science Chinese Academy of Sciences Hefei China

In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As m → 0 and m → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.

关键词： Auxiliary Equation Method nonlinear evolution equations Periodic Wave Solutions Mapping Method Solitary Wave Solutions

Solving second-order nonlinear evolution partial differential equations using deep learning

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Communications in Theoretical Physics 2020年第10期72卷 42-52页

Solving nonlinear evolution partial differential equations has been a longstanding computational *** this paper,we present a universal paradigm of learning the system and extracting patterns from data generated from ***,this framework approximates the latent solution with a deep neural network,which is trained with the constraint of underlying physical laws usually expressed by some *** particular,we test the effectiveness of the approach for the Burgers'equation used as an example of second-order nonlinear evolution equations under different initial and boundary *** results also indicate that for soliton solutions,the model training costs significantly less time than other initial conditions.

关键词： deep learning nonlinear evolution equations data-driven solutions solitons nonlinear dynamics

Lie symmetry analysis and invariant solutions for(2+1) dimensional Bogoyavlensky-Konopelchenko equation with variable-coefficient in wave propagation

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Journal of Ocean Engineering and Science 2022年第3期7卷 248-254页

作者： Mohamed R.Ali Wen-Xiu Ma R.Sadat Department of Basic Science Faculty of Engineering at BenhaBenha UniversityBenhaEgypt Department of Mathematics Zhejiang Normal UniversityJinhua 321004ZhejiangChina Department of Mathematics King Abdulaziz UniversityJeddah 21589Saudi Arabia Department of Mathematics and Statistics University of South FloridaTampaFL 33620-5700USA School of Mathematics South China University of TechnologyGuangzhou 510640China Department of Mathematics Zagazig Faculty of EngineeringZagazig UniversityZagazigEgypt

This work aims to present nonlinear models that arise in ocean *** are many models of ocean waves that are present in *** shallow water,the linearization of the equations requires critical conditions on wave capacity than it make in deep water,and the strong nonlinear belongings are *** use Lie symmetry analysis to obtain different types of soliton solutions like one,two,and three-soliton solutions in a(2+1)dimensional variable-coefficient Bogoyavlensky Konopelchenko(VCBK)equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and *** use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK *** demonstrate the physical meaning of the solutions obtained by the presented techniques,the graphical performance has been demonstrated with some *** presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.

关键词： Symmetry approach Solitons Partial differential equations The variable coefficients(2+1)-dimensional Bogoyavlensky Konopelchenko equation nonlinear evolution equations

Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

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Chinese Physics B 2011年第1期20卷 42-49页

作者：朱春蓉 College of Mathematics and Computer Science Anhui Normal University

The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full. description, of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invaxiant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite

关键词： nonlinear evolution equations cubic operators invariant subspace method submaximal dimension blow-up solution

A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model

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Chinese Physics B 2008年第2期17卷 399-402页

作者：潘军廷龚伦训 School of Science Guizhou Normal University

Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony （mBBM） model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations.

关键词： nonlinear evolution equations new expansion method mBBM model exact solutions